Problem: $\dfrac{dy}{dx}=-4y$, and $y=3$ when $x=2$. Solve the equation. Choose 1 answer: Choose 1 answer: (Choice A) A $y=6e^{-4x}$ (Choice B) B $y=3e^{8-4x}$ (Choice C) C $y=3e^{-4x}$ (Choice D) D $y=3e^{4-4x}$
Solution: The general solution of equations of the form $\dfrac{dy}{dx}=ky$ is $y=C\cdot e^{kx}$ for some constant $C$. This can be found using separation of variables. In our case, $k=-4$, so $y=C\cdot e^{-4x}$. Let's use the fact that $y=3$ when $x=2$ to find $C$ : $\begin{aligned} y&=C\cdot e^{-4x} \\\\ 3&=C\cdot e^{-4\cdot 2} \gray{\text{Plug }x=2\text{ and }y=3} \\\\ 3e^{8}&=C \end{aligned}$ In conclusion, $y=3e^{8-4x}$.